Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 32

Solve the inequality $|5 - x| > 3$, and graph the solution set.

The absolute value inequality is rewritten as

$5-x > 3$ or $5-x < -3$,

because $5 - x$ must represent a number that is more than $3$ units from on either side of the number line. We can solve the compound inequality.


$
\begin{equation}
\begin{aligned}

|5 - x| > & 3 && \qquad \text{or} &&& 5 - x < & -3
&&
\\
-x > & -2 && \qquad \text{or} &&& -x < & -8
&& \text{Subtract } 5
\\
x < & 2 && \qquad \text{or} &&& x > & 8
&& \text{Divide $-1$. Reverse the inequality symbols.}

\end{aligned}
\end{equation}
$


The solution set is $(- \infty, 2) \cup (8, \infty)$.